2-Approximating Feedback Vertex Set in Tournaments

Lokshtanov, Daniel; Misra, Pranabendu; Mukherjee, Joydeep; Panolan, Fahad; Philip, Geevarghese; Saurabh, Saket

doi:10.1137/1.9781611975994.61

Cited by 5 publications

(3 citation statements)

References 26 publications

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“…Recently, for every > 0, a factor (2 + )-approximation algorithm for deleting vertices to get a split graph has been obtained [39]. However, for our purposes Observation 2.2 will suffice.…”

Section: Observation 22mentioning

confidence: 96%

On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

Gunda

Jain

Lokshtanov

et al. 2021

ACM Trans. Comput. Theory

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A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting k edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely, the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this article, we study the F -Contraction problem, where F is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph G and an integer k , F -Contraction asks whether there exists X ⊆ E(G) such that G/X ∈ F and | X | ≤ k . Here, G/X is the graph obtained from G by contracting edges in X . We obtain the following results for the F - Contraction problem: • Clique Contraction is known to be FPT . However, unless NP⊆ coNP/ poly , it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme ( PSAKS ). That is, it admits a (1 + ε)-approximate kernel with O ( k f(ε)) vertices for every ε > 0. • Split Contraction is known to be W[1]-Hard . We deconstruct this intractability result in two ways. First, we give a (2+ε)-approximate polynomial kernel for Split Contraction (which also implies a factor (2+ε)- FPT -approximation algorithm for Split Contraction ). Furthermore, we show that, assuming Gap-ETH , there is no (5/4-δ)- FPT -approximation algorithm for Split Contraction . Here, ε, δ > 0 are fixed constants. • Chordal Contraction is known to be W[2]-Hard . We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming FPT ≠ W[1] , there is no F(k) - FPT -approximation algorithm for Chordal Contraction . Here, F(k) is an arbitrary function depending on k alone. We say that an algorithm is an h(k) - FPT -approximation algorithm for the F -Contraction problem, if it runs in FPT time, and on any input (G, k) such that there exists X ⊆ E(G) satisfying G/X ∈ F and | X | ≤ k , it outputs an edge set Y of size at most h(k) ċ k for which G/Y is in F .

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“…Recently, for every > 0, a factor (2 + )-approximation algorithm for deleting vertices to get a split graph has been obtained [39]. However, for our purposes Observation 2.2 will suffice.…”

Section: Observation 22mentioning

confidence: 96%

On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

Gunda

Jain

Lokshtanov

et al. 2021

ACM Trans. Comput. Theory

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“…Cai, Deng, and Zang [8] gave a 5/2-approximation algorithm for FVST, which was later improved to a 7/3-approximation algorithm by Mnich, Williams, and Végh [19]. Lokshtanov, Misra, Mukherjee, Panolan, Philip, and Saurabh [18] recently gave a randomized 2-approximation algorithm, but no deterministic (polynomial-time) 2-approximation algorithm is known. For FVST, one round of the Sherali-Adams hierarchy actually provides a 7/3-approximation [3].…”

Section: Other Related Workmentioning

confidence: 99%

A tight approximation algorithm for the cluster vertex deletion problem

et al. 2022

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We give the first 2-approximation algorithm for the cluster vertex deletion problem. This is tight, since approximating the problem within any constant factor smaller than 2 is UGC-hard. Our algorithm combines the previous approaches, based on the local ratio technique and the management of true twins, with a novel construction of a "good" cost function on the vertices at distance at most 2 from any vertex of the input graph.As an additional contribution, we also study cluster vertex deletion from the polyhedral perspective, where we prove almost matching upper and lower bounds on how well linear programming relaxations can approximate the problem.

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“…By a well known result of Lewis and Yannakakis [56] almost all Π-Deletion problems are NP-complete. For this reason the study of such problems has mostly been from the perspective of methods for coping with computational intractability, such as approximation [3,4,5,11,25,31,35,42,52,58,59,62,76,77,79,80], exact [7,29,33,28,30,78], or parameterized algorithms [10,12,16,26,27,31,44,54,63,65,57,21,24]. In this paper we focus on parameterized algorithms for Π-Deletion problems: more concretely, for every property Π the aim is to design an algorithm for Π-Deletion that given a graph G and integer k, determines in time f (k)n O (1) time whether a solution set S of size at most k exists.…”

Section: Introductionmentioning

confidence: 99%

Hitting Topological Minors is FPT

Fomin¹,

Lokshtanov²,

Panolan³

et al. 2019

Preprint

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In the Topological Minor Deletion (TM-Deletion) problem input consists of an undirected graph G, a family of undirected graphs F and an integer k. The task is to determine whether G contains a set of vertices S of size at most k, such that the graph G \ S obtained from G by removing the vertices of S, contains no graph from F as a topological minor. We give an algorithm for TM-Deletion with running time f (h , k) • |V (G)| 4 . Here h is the maximum size of a graph in F and f is a computable function of h and k. This is the first fixed parameter tractable algorithm (FPT) for the problem. In fact, even for the restricted case of planar inputs the first FPT algorithm was found only recently by Golovach et al. [SODA 2020]. For this case we improve upon the algorithm of Golovach et al. [SODA 2020] by designing an FPT algorithm with explicit dependence on k and h .

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2-Approximating Feedback Vertex Set in Tournaments

Cited by 5 publications

References 26 publications

On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

A tight approximation algorithm for the cluster vertex deletion problem

Hitting Topological Minors is FPT

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